Why Eisenstein Proved the Eisenstein Criterion and Why Schönemann Discovered It First

In this paper the author not only tells us that Eisenstein was scooped by Theodor Schönemann but, much more interestingly and importantly, he explains why both men were led to the result. It's an engrossing tale beginning with Gauss's equal division of the circle, the relation of that work to the analogous problem on a lemniscate, the connection of that problem to the question of solvability by radicals of polynomials, and thence into Galois theory, finite fields, and elliptic curves. It is a rich story, not of a priority dispute but of a sweeping flow of ideas beginning with Gauss (who partially scooped both Schönemann and Eisenstein) and extending into the heart of modern-day mathematics.

David A. Cox went to Rice University and received his Ph.D. from Princeton University in 1975. After teaching at Haverford and Rutgers, he has been at Amherst College since 1979, except for a sabbatical at Oklahoma State University. After more than 30 years, he still loves the combination of teaching and scholarship that is possible at a liberal arts college. His current areas of research include toric varieties and the commutative algebra of curve parametrizations. His earlier work in algebraic geometry includes papers on étale homotopy theory, elliptic surfaces, and infinitesimal variations of Hodge structure, and he also has interests in number theory and the history of mathematics. He is the author of books on number theory, computational algebraic geometry, mirror symmetry, Galois theory, and toric varieties, three of which have been translated into Japanese.

In this paper the author not only tells us that Eisenstein was scooped by Theodor Schönemann but, much more interestingly and importantly, he explains why both men were led to the result. It’s an engrossing tale beginning with Gauss’s equal division of the circle, the relation of that work to the analogous problem on a lemniscate, the connection of that problem to the question of solvability by radicals of polynomials, and thence into Galois theory, finite fields, and elliptic curves. It is a rich story, not of a priority dispute but of a sweeping flow of ideas beginning with Gauss (who partially scooped both Schönemann and Eisenstein) and extending into the heart of modern-day mathematics.